Challenges in Geometric Numerical Integration

Geometric Numerical Integration is a subfield of the numerical treatment of differential equations. It deals with the design and analysis of algorithms that preserve the structure of the analytic flow. The present review discusses numerical integrators, which nearly preserve the energy of Hamiltonian systems over long times. Backward error analysis gives important insight in the situation, where the product of the step size with the highest frequency is small. Modulated Fourier expansions permit to treat nonlinearly perturbed fast oscillators. A big challenge that remains is to get insight into the long-time behavior of numerical integrators for fully nonlinear oscillatory problems, where the product of the step size with the highest frequency is not small. 1 Geometric Numerical Integration Ordinary differential equations arise everywhere in science and their numerical treatment is of great importance. The development took place in three periods: the numerical solution of non-stiff differential equations started in the end of the 19th century, whereas that of stiff problems began in the middle of the 20th century and was motivated by space discretizations of parabolic differential equations and by simulations of chemical reactions. With the interest in computations over long time intervals one discovered that certain methods reproduce the qualitative behavior of the exact flow much better than others. In the late 1980ties numerical analysts started to design and study (we quote from the preface of the monograph [9]) . . . numerical methods that preserve geometric properties of the flow of a differential equation: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, . . . and methods for problems with highly oscillatory solutions. Ernst Hairer Sect. de mathématiques, 2-4 rue du Lièvre, Univ. de Genève, CH-1211 Genève 4, Switzerland, e-mail: [email protected]